Maximal quasi-normed extension of quasi-normed lattices

The purpose of this article is to extend the Abramovich's construction of a maximal normed extension of a normed lattice to quasi-Banach setting. It is proved that the maximal quasi-normed extension $X^\varkappa$ of a Dedekind complete quasi-normed lattice $X$ with the weak $\sigma$-Fatou property is a quasi-Banach lattice if and only if $X$ is intervally complete. Moreover, $X^\varkappa$ has the Fatou and the Levi property provided that $X$ is a Dedekind complete quasi-normed space with the Fatou property. The possibility of applying this construction to the definition of a space of weakly integrable functions with respect to a measure taking values from a quasi-Banach lattice is also discussed, since the duality based definition does not work in the quasi-Banach setting.